Transcript Strategic Dynamics: Introduction to Game Theory

Introduction to Game Theory
Networked Life
CSE 112
Spring 2005
Prof. Michael Kearns
Game Theory
• A mathematical theory designed to model:
– how rational individuals should behave
– when individual outcomes are determined by collective behavior
– strategic behavior
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Rational usually means selfish --- but not always
Rich history, flourished during the Cold War
Traditionally viewed as a subject of economics
Subsequently applied by many fields
– evolutionary biology, social psychology
• Perhaps the branch of pure math most widely examined
outside of the “hard” sciences
Prisoner’s Dilemma
cooperate defect
cooperate -1, -1
defect
-10, -0.25
-0.25, -10 -8, -8
• Cooperate = deny the crime; defect = confess guilt of both
• Claim that (defect, defect) is an equilibrium:
– if I am definitely going to defect, you choose between -10 and -8
– so you will also defect
– same logic applies to me
• Note unilateral nature of equilibrium:
– I fix a behavior or strategy for you, then choose my best response
• Claim: no other pair of strategies is an equilibrium
• But we would have been so much better off cooperating…
Penny Matching
heads
tails
heads
1, 0
0, 1
tails
0, 1
1, 0
• What are the equilibrium strategies now?
• There are none!
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if I play heads then you will of course play tails
but that makes me want to play tails too
which in turn makes you want to play heads
etc. etc. etc.
• But what if we can each (privately) flip coins?
– the strategy pair (1/2, 1/2) is an equilibrium
• Such randomized strategies are called mixed strategies
The World According to Nash
• If > 2 actions, mixed strategy is a distribution on them
– e.g. 1/3 rock, 1/3 paper, 1/3 scissors
• Might also have > 2 players
• A general mixed strategy is a vector P = (P[1], P[2],… P[n]):
– P[i] is a distribution over the actions for player i
– assume everyone knows all the distributions P[j]
– but the “coin flips” used to select from P[i] known only to i
• P is an equilibrium if:
– for every i, P[i] is a best response to all the other P[j]
• Nash 1950: every game has a mixed strategy equilibrium
– no matter how many rows and columns there are
– in fact, no matter how many players there are
• Thus known as a Nash equilibrium
• A major reason for Nash’s Nobel Prize in economics
Facts about Nash Equilibria
• While there is always at least one, there might be many
– zero-sum games: all equilibria give the same payoffs to each player
– non zero-sum: different equilibria may give different payoffs!
• Equilibrium is a static notion
– does not suggest how players might learn to play equilibrium
– does not suggest how we might choose among multiple equilibria
• Nash equilibrium is a strictly competitive notion
– players cannot have “pre-play communication”
– bargains, side payments, threats, collusions, etc. not allowed
• Computing Nash equilibria for large games is difficult
Hawks and Doves
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hawk
dove
hawk
(V-C)/2, (V-C)/2
V, 0
dove
0, V
V/2, V/2
Two parties confront over a resource of value V
May simply display aggression, or actually have a fight
Cost of losing a fight: C > V
Assume parties are equally likely to win or lose
There are three Nash equilibria:
– (hawk, dove), (dove, hawk) and (V/C hawk, V/C hawk)
• Alternative interpretation for C >> V:
– the Kansas Cornfield Intersection game (a.k.a. Chicken)
– hawk = speed through intersection, dove = yield
Board Games and Game Theory
• What does game theory say about richer games?
– tic-tac-toe, checkers, backgammon, go,…
– these are all games of complete information with state
– incomplete information: poker
• Imagine an absurdly large “game matrix” for chess:
– each row/column represents a complete strategy for playing
– strategy = a mapping from every possible board configuration to the
next move for the player
– number of rows or columns is huge --- but finite!
• Thus, a Nash equilibrium for chess exists!
– it’s just completely infeasible to compute it
– note: can often “push” randomization “inside” the strategy
Repeated Games
• Nash equilibrium analyzes “one-shot” games
– we meet for the first time, play once, and separate forever
• Natural extension: repeated games
– we play the same game (e.g. Prisoner’s Dilemma) many times in a row
– like a board game, where the “state” is the history of play so far
– strategy = a mapping from the history so far to your next move
• So repeated games also have a Nash equilibrium
– may be different from the one-shot equilibrium!
– depends on the game and details of the setting
• We are approaching learning in games
– natural to adapt your behavior (strategy) based on play so far
Repeated Prisoner’s Dilemma
• If we play for R rounds, and both know R:
– (always defect, always defect) still the only Nash equilibrium
– argue by backwards induction
• If uncertainty about R is introduced (e.g. random stopping):
– cooperation and tit-for-tat can become equilibria
• If computational restrictions are placed on our strategies:
– as long as we’re too feeble to count, cooperative equilibria arise
– formally: < log(R) states in a finite automaton
– a form of bounded rationality
The Folk Theorem
• Take any one-shot, two-player game
• Suppose that (u,v) are the (expected) payoffs under some
mixed strategy pair (P[1],P[2]) for the two players
– (P[1], P[2]) not necessarily a Nash equilibrium
– but (u,v) gives better payoffs than the security levels
• security level: what a player can get no matter what the other does
– example: sec. level is (-8, -8) in Prisoner’s Dilemma; (-1,-1) is better
• Then there is always a Nash equilibrium for the infinite
repeated game giving payoffs (u,v)
– makes use of the concept of threats
• Partial resolution of the difficulties of Nash equilibria…
Correlated Equilibrium
• In a Nash equilibrium (P[1],P[2]):
– player 2 “knows” the distribution P[1]
– but doesn’t know the “random bits” player 1 uses to select from P[1]
– equilibrium relies on private randomization
• Suppose now we also allow public (shared) randomization
– so strategy might say things like “if private bits = 100110 and
shared bits = 110100110, then play hawk”
• Then two strategies are in correlated equilibrium if:
– knowing only your strategy and the shared bits, my strategy is a
best response, and vice-versa
• Nash is the special case of no shared bits
Hawks and Doves Revisited
hawk
dove
hawk
(V-C)/2, (V-C)/2
V, 0
dove
0, V
V/2, V/2
• There are three Nash equilibria:
– (hawk, dove), (dove, hawk) and (V/C hawk, V/C hawk)
• Alternative interpretation for C >> V:
– the Kansas Cornfield Intersection game (a.k.a. Chicken)
– hawk = speed through intersection, dove = yield
• Correlated equilibrium: the traffic signal
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if the shared bit is green to me, I am playing hawk
if the shared bit is red to me, I will play dove
you play the symmetric strategy
splits waiting time between us --- a different outcome than Nash
Correlated Equilibrium Facts
• Always exists
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all Nash equilibria are correlated equilibria
all probability distributions over Nash equilibria are C.E.
and some more things are C.E. as well
a broader concept than Nash
• Technical advantages of correlated equilibria:
– often easier to compute than Nash
• Conceptual advantages:
– correlated behavior is a fact of the real world
– model a limited form of cooperation
– more general cooperation becomes extremely complex and messy
• Breaking news (late 90s – now):
– CE is the natural convergence notion for “rational” learning in games!
Next Up
• Have so far examined simple games between two players
• Strategic interaction on the smallest “network”:
– two vertices with a single link between them
– much richer interaction than just info transmission, messages, etc.
• Classical game theory generalizes to many players
– e.g. Nash equilibria always exist in multi-player matrix games
– but this fails to capture/exploit/examine structured interaction
• We need specific models for networked games:
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games on networks: local interaction
shared information: economies, financial markets
voting systems
evolutionary games